{"id":2082,"date":"2012-06-25T09:00:09","date_gmt":"2012-06-25T16:00:09","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=2082"},"modified":"2020-01-15T10:50:39","modified_gmt":"2020-01-15T18:50:39","slug":"fractions-on-the-gmat","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/fractions-on-the-gmat\/","title":{"rendered":"Fractions on the GMAT!"},"content":{"rendered":"

Learn to master this often confusing topic!\u00a0<\/strong><\/p>\n

Why Fractions Are Hard<\/h2>\n

Of all the math topics that raise dread, fear, anxiety, and confusion, few do so as consistently and as potently as do fractions.\u00a0 I have my own personal theory why fractions are hard.\u00a0 The trouble is: think about when you learned fractions — maybe the third, fourth, and\/or fifth grades.\u00a0 That’s when fractions are usually taught, but there are two problems with that.\u00a0 First of all, that’s before the tsunami of puberty hits you and virtually obliterated all previously held logical connections in your head.\u00a0 More importantly, fractions, like many other topics in math, involve sophisticated patterns, but in the fourth grade, no one is capable of abstraction, so instead you are just taught to reproduce patterns mechanically, and relying on mechanical repetition has severe limitations: similarly looking things become conflated, and when you get confused, you basically don’t know what to do.\u00a0 Many people simply give up at that point.<\/p>\n

The solution is re-approach those mechanical procedures, but with understanding<\/em>.\u00a0 When you understand why you do each thing, then (a) you can remember it much better, and (b) in a moment of confusion you can figure out what to do.\u00a0 I will lead you through fractions from the ground up.<\/p>\n

 <\/p>\n

What is a Fraction?<\/h2>\n

A fraction is a way of showing division.\u00a0 The fraction 2\/7 means the number you get when you divide 2 by 7.\u00a0\u00a0 The top of a fraction is called the numerator<\/strong>, and the bottom of a fraction is called the denominator<\/strong>.<\/p>\n

The fraction 2\/7 also means the following.\u00a0 Imagine dividing something whole into seven equal parts—one of those parts is 1\/7 of the whole, so 2\/7 = 2*(1\/7) is two of those parts.\u00a0 Probably this diagram will call up dim memories from your prepubescent mathematical experiences.<\/p>\n

\"\"<\/a><\/p>\n

That, visually, represents 2\/7.\u00a0 Two vital things to remember about 2\/7—one is that 2\/7 is two of the thing called 1\/7—that is, 2\/7 = 2*(1\/7), and second is this visual perspective that is always vital.<\/p>\n

Notice that if you have the fractions 4\/14 or 10\/35, they both cancel down to 2\/7.\u00a0 Cancelling is division<\/span><\/strong>.\u00a0 That’s a big idea—thus, when you have 4\/20, and you cancel (i.e. divide) the 4’s in the numerator and the denominator, they don’t simply “go away” (a fourth-grade mechanical way of thinking), but rather what’s left in the numerator is 4 \u00f7 4 = 1, and we get 4\/20 = 1\/5.<\/p>\n

 <\/p>\n

Adding and Subtracting Fractions<\/h2>\n

First of all, let’s address the common mistake: when you add fractions, you can’t simply add across in the numerator and denominator (this is the mistake people make when they mechanically perform the rule for multiplication with addition instead!)<\/p>\n

\"\"<\/a><\/p>\n

You may dimly remember that you only can add and subtract fractions when you have a common denominator<\/span><\/strong>.\u00a0 That’s true, but why is that true?\u00a0 Believe it or not, the basis of this fact is none other than the Distributive Law, a(b + c) = ab + ac.\u00a0 For example, if I add 3x and 5x, I get 3x + 5x = 8x — according to the Distributive Law, I can add two terms of the same thing, BUT if I want to add 3x + 5y, I can’t simplify that any further and must keep it as 3x + 5y.\u00a0 If you add two terms of the same underlying thing, you can combine the terms, but if you are adding proverbial apples and oranges, you can’t combine.\u00a0 Well, 3\/11 + 5\/11 is really just 3*(1\/11) + 5*(1\/11) — so, by the Distributive Law, you are allowed to add two terms of the same thing: 3\/11 + 5\/11 = 8\/11<\/p>\n

When the denominators are not the same — 3\/8 + 1\/6 — then you can’t add them as is, but you can take advantage of a sleek mathematical trick.\u00a0 We know that any number over itself, say 3\/3, equals 1, and you can always multiply by 1 and not change the value of something.\u00a0 Therefore, I could multiply 3\/8 by some a\/a, and multiply 1\/6 by some other b\/b, and both would retain the same value.\u00a0 I want to multiply each so that I find the Least Common Denominator<\/a>\u00a0 (LCD), which here is 24.\u00a0 Thus<\/p>\n

\"\"<\/a><\/p>\n

The same thing works for subtraction:<\/p>\n

\"\"<\/a><\/p>\n

 <\/p>\n

Multiplying Fractions<\/h2>\n

This is the easiest of all fractions rules.\u00a0 To multiply fractions, multiply across in the numerators and denominators.<\/p>\n

\"\"<\/a><\/p>\n

What’s a little tricky about multiply is what you can cancel.\u00a0 If you are multiply two fractions, of course you can cancel any numerator with its own denominator, but you can also cancel one numerator with another denominator.\u00a0 Sometimes, that is called “cross-cancelling”, which I think is a 100% useless term that reinforces fourth-grade mechanical thinking.\u00a0 I think it’s much more effective to remember: when you multiply fractions, you can cancel any numerator with any denominator<\/strong>.\u00a0 Here’s a horrendous multiplication problem that simplifies elegantly with the liberal use of cancelling.<\/p>\n

\"\"<\/a><\/p>\n

 <\/p>\n

Dividing Fractions<\/h2>\n

First of all, multiplying by 1\/3 is the same as dividing by 3.\u00a0 That’s just the fundamental definition of fraction as division.\u00a0 This also means, dividing by 1\/3 is the same as multiplying by 3.\u00a0 This pattern suggests, correctly, that dividing by a fraction simply means multiplying by its reciprocal:<\/p>\n

\"\"<\/a><\/p>\n

Notice, as always, cancel before<\/span><\/em> you multiply.\u00a0 Dividing a fraction by a number follows the same pattern:<\/p>\n

\"\"<\/a><\/p>\n

Notice, this is really the same idea as: dividing by 3 means the same thin as multiplying by 1\/3.\u00a0 Also, again, notice we cancel before<\/span><\/em> we multiply.<\/p>\n

 <\/p>\n

Proportions<\/h2>\n

Another word for a fraction is a ratio<\/strong>: ratios and fractions are exactly the same thing.\u00a0 A proportion is when you have two ratios, two fractions, set equal to each other.\u00a0 For example,<\/p>\n

\"\"<\/a><\/p>\n

One legitimate move is to cross-multiply<\/strong>, although doing so here would violate the ultra-strategic dictum: cancel before<\/span><\/em> you multiply.\u00a0 And it’s precisely this issue, what can you cancel and what can’t cancel in a proportion, that causes endless confusion.\u00a0 Let’s look at the general proportion a\/b = c\/d.<\/p>\n

First of all, as always, you can cancel any numerator with its own denominator — you can cancel common factors in a & b, or in c & d.\u00a0 Furthermore, a proportion, by its very nature, is an equation, and you can always multiply or divide both sides of an equation by the same thing.\u00a0 This means: you can cancel common factors in both numerators (a & c) or in both denominators (b & d).\u00a0 The following diagrams summarize all the legitimate directions of cancellation in a proportion.<\/p>\n

\"\"<\/a>\"\"<\/a><\/p>\n

The following are highly tempting but complete illegal ways to cancel in proportions:<\/p>\n

\"\"<\/a>\"\"<\/a><\/p>\n

The trouble is, folks mechanically memorize the cancelling pattern for multiplying fractions — or even worse, they learn an utterly useless term like “cross-cancelling” — and then they mechanically apply that pattern when there’s an equal sign between the two fractions instead of a multiplication sign.\u00a0 This is a major mistake, and any time a proportion appears on the GMAT, the test-maker is expecting a large flock of test-takers to fall into this trap.<\/p>\n

Let’s solve the proportion we wrote above, with proper cancelling:<\/p>\n

\"\"<\/a><\/p>\n

Notice, in that last step, to isolate x, all we had to do was multiply both sides by 3.\u00a0 Cross-multiplying, while always legal in a proportion, often is a waste of time that simply adds extra steps.<\/p>\n

 <\/p>\n

Practice Questions<\/h2>\n

I hope this refresher has clear up some fractions concepts for you.\u00a0 The best way to cement a new mathematical understand: practice, practice, practice!<\/p>\n

1) https:\/\/gmat.magoosh.com\/questions\/124<\/a><\/p>\n

2) https:\/\/gmat.magoosh.com\/questions\/812<\/a><\/p>\n

3) https:\/\/gmat.magoosh.com\/questions\/62<\/a><\/p>\n

 <\/p>\n

Special Note:<\/h4>\n

To find out where fractions sit in the “big picture” of GMAT Quant, and what other Quant concepts you should study, check out our post entitled:<\/p>\n

What Kind of Math is on the GMAT? Breakdown of Quant Concepts by Frequency<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Learn to master this often confusing topic!\u00a0 Why Fractions Are Hard Of all the math topics that raise dread, fear, anxiety, and confusion, few do so as consistently and as potently as do fractions.\u00a0 I have my own personal theory why fractions are hard.\u00a0 The trouble is: think about when you learned fractions — maybe […]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[150],"tags":[],"ppma_author":[13209],"class_list":["post-2082","post","type-post","status-publish","format-standard","hentry","category-basics"],"acf":[],"yoast_head":"\nFractions on the GMAT! - Magoosh Blog \u2014 GMAT\u00ae Exam<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/magoosh.com\/gmat\/fractions-on-the-gmat\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Fractions on the GMAT!\" \/>\n<meta property=\"og:description\" content=\"Learn to master this often confusing topic!\u00a0 Why Fractions Are Hard Of all the math topics that raise dread, fear, anxiety, and confusion, few do so as consistently and as potently as do fractions.\u00a0 I have my own personal theory why fractions are hard.\u00a0 The trouble is: think about when you learned fractions — maybe […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/magoosh.com\/gmat\/fractions-on-the-gmat\/\" \/>\n<meta property=\"og:site_name\" content=\"Magoosh Blog \u2014 GMAT\u00ae Exam\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/MagooshGMAT\/\" \/>\n<meta property=\"article:published_time\" content=\"2012-06-25T16:00:09+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2020-01-15T18:50:39+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/magoosh.com\/gmat\/files\/2012\/06\/fractions_img1.png\" \/>\n<meta name=\"author\" content=\"Mike M\u1d9cGarry\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@MagooshGMAT\" \/>\n<meta name=\"twitter:site\" content=\"@MagooshGMAT\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Mike M\u1d9cGarry\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"6 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/magoosh.com\/gmat\/fractions-on-the-gmat\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/fractions-on-the-gmat\/\"},\"author\":{\"name\":\"Mike M\u1d9cGarry\",\"@id\":\"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/320346c205075513344435baf9b0521b\"},\"headline\":\"Fractions on the GMAT!\",\"datePublished\":\"2012-06-25T16:00:09+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/fractions-on-the-gmat\/\"},\"wordCount\":1282,\"commentCount\":13,\"publisher\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/#organization\"},\"articleSection\":[\"GMAT Math Basics\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/magoosh.com\/gmat\/fractions-on-the-gmat\/\",\"url\":\"https:\/\/magoosh.com\/gmat\/fractions-on-the-gmat\/\",\"name\":\"Fractions on the GMAT! 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Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations.","sameAs":["https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured"],"award":["Magna cum laude from Harvard"],"knowsAbout":["GMAT"],"knowsLanguage":["English"],"jobTitle":"Content Creator","worksFor":"Magoosh","url":"https:\/\/magoosh.com\/gmat\/author\/mikemcgarry\/"}]}},"authors":[{"term_id":13209,"user_id":26,"is_guest":0,"slug":"mikemcgarry","display_name":"Mike M\u1d9cGarry","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","user_url":"","last_name":"M\u1d9cGarry","first_name":"Mike","description":"Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as \"member of the month\" for over two years at <a href=\"https:\/\/gmatclub.com\/blog\/2012\/09\/mike-mcgarrys-gmat-experience\/\" rel=\"noopener noreferrer\">GMAT Club<\/a>. Mike holds an A.B. in Physics (graduating <em>magna cum laude<\/em>) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's <a href=\"https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured\" rel=\"noopener noreferrer\">Youtube <\/a>video explanations and resources like <a href=\"https:\/\/magoosh.com\/gmat\/whats-a-good-gmat-score\/\" rel=\"noopener noreferrer\">What is a Good GMAT Score?<\/a> and the <a href=\"https:\/\/magoosh.com\/gmat\/gmat-diagnostic-test\/\" rel=\"noopener noreferrer\">GMAT Diagnostic Test<\/a>."}],"_links":{"self":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/2082","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/users\/26"}],"replies":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/comments?post=2082"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/2082\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/media?parent=2082"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/categories?post=2082"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/tags?post=2082"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/ppma_author?post=2082"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}